Problem 25 - College Algebra with Modeling and Visualization

Question

Simplify. $$ \frac{3}{7}-\frac{4}{7} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{-1}{7} \).

Step 1: Identify Like Terms

When subtracting fractions with the same denominator, you can directly subtract the numerators. In the expression \( \frac{3}{7} - \frac{4}{7} \), both fractions have the same denominator, which is 7.

Step 2: Subtract the Numerators

Since the denominators are the same, subtract the numerators: \( 3 - 4 \). This gives us a numerator of \( -1 \).

Step 3: Write the Result as a Fraction

Combine the result from the numerator with the common denominator. This results in the fraction \( \frac{-1}{7} \).

Key Concepts

Subtracting Fractions

Subtracting fractions involves a few simple steps that become easier with practice. The key is to pay attention to the numerators and denominators.
First, when fractions have identical denominators, like the fractions \( \frac{3}{7} \) and \( \frac{4}{7} \) in our example, you can subtract the numerators directly.
You perform the subtraction on the top numbers (the numerators). Here, \( 3 - 4 = -1 \). It’s important to line up the numerators correctly and subtract them.
If the denominators are different, however, you must find a common denominator before subtracting, but we'll discuss that further in the next section!

Common Denominator

A common denominator is essential when dealing with fractions that don’t initially share the same denominator. For the current problem, both fractions already have 7 as their denominator, allowing for direct subtraction.
However, if you encounter fractions like \( \frac{2}{6} \) and \( \frac{1}{4} \), you’ll need to convert them so they have a common denominator before subtracting.

• Find the least common multiple (LCM) of the denominators.
• Adjust each fraction, multiplying the numerator and denominator by whatever number makes the denominator equal to the LCM.

Once you’ve transformed the fractions to have this shared denominator, you can easily subtract the numerators as shown in the previous section.

Negative Fractions

Sometimes when subtracting fractions, as shown in our example, you end up with a negative numerator. This is completely okay, and it simply results in a negative fraction.
In our example, you subtracted and got \( -1 \) as the numerator, leading to \( \frac{-1}{7} \). A negative fraction like this can represent a loss or a reduction in quantity.
Consider how to handle negative signs:

• If the numerator is negative, place the negative sign in front of the fraction: \( \frac{-1}{7} \).
• If you prefer, you can write it with the negative sign before the whole fraction. In arithmetic terms, \( \frac{-1}{7} \) and \( -\frac{1}{7} \) are equivalent.

Understanding negative fractions will help in various math scenarios, including subtracting larger fractions and working through algebraic expressions.